Alpha Question Bank for Engineering: Mathematics>Chapter 9 - Matrices and Determinants>Exercise-2>Q 41

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Suppose $a_{1}, a_{2}, a_{3}$ are in A.P. and $b_{1}, b_{2}, b_{3}$ are in H.P. and let $Δ = |\begin{array}{l} a_{1} - b_{1} & a_{1} - b_{2} & a_{1} - b_{3} \\ a_{2} - b_{1} & a_{2} - b_{2} & a_{2} - b_{3} \\ a_{3} - b_{1} & a_{3} - b_{2} & a_{3} - b_{3} \end{array}|,$ then

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Important Questions on Matrices and Determinants

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Alpha Question Bank for Engineering: Mathematics>Chapter 9 - Matrices and Determinants>Exercise-2>Q 42

Let

θ = \frac{π}{5}, X = [\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix}], O

is null matrix and

I

is an identity matrix of order

2 \times 2,

and if

I + X + X^{2} + \dots \dots + X^{n} = O,

then

n

can be

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Alpha Question Bank for Engineering: Mathematics>Chapter 9 - Matrices and Determinants>Exercise-2>Q 43

If

Δ = |\begin{matrix} x & 2 y - z & - z \\ y & 2 x - z & - z \\ y & 2 y - z & 2 x - 2 y - z \end{matrix}|,

then

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Alpha Question Bank for Engineering: Mathematics>Chapter 9 - Matrices and Determinants>Exercise-2>Q 44

Let

a, b > 0

and

Δ = |\begin{matrix} - x & a & b \\ b & - x & a \\ a & b & - x \end{matrix}|,

then

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Alpha Question Bank for Engineering: Mathematics>Chapter 9 - Matrices and Determinants>Exercise-2>Q 45

The determinant

Δ = |\begin{matrix} b & c & b α + c \\ c & d & c α + d \\ b α + c & c α + d & a α^{3} - c α \end{matrix}|

is equal to zero, if

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Alpha Question Bank for Engineering: Mathematics>Chapter 9 - Matrices and Determinants>Exercise-2>Q 46

The determinant

Δ = |\begin{matrix} a^{2} (1 + x) & a b & a c \\ a b & b^{2} (1 + x) & b c \\ a c & b c & c^{2} (1 + x) \end{matrix}|

is divisible by

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Alpha Question Bank for Engineering: Mathematics>Chapter 9 - Matrices and Determinants>Exercise-2>Q 47

If

A

is a non-singular matrix and

A^{T}

denotes the transpose of

A,

then:

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Alpha Question Bank for Engineering: Mathematics>Chapter 9 - Matrices and Determinants>Exercise-2>Q 48

Let

f (x) = |\begin{matrix} 2 \sin x & \sin^{2} x & 0 \\ 1 & 2 \sin x & \sin^{2} x \\ 0 & 1 & 2 \sin x \end{matrix}|,

then

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Alpha Question Bank for Engineering: Mathematics>Chapter 9 - Matrices and Determinants>Exercise-2>Q 49

Let

Δ = |\begin{matrix} 1 & x & x^{2} \\ x^{2} & 1 & x \\ x & x^{2} & 1 \end{matrix}|,

then